Binary signals: a note on the prime period of a point

Abstract

The ’nice’ x:R→{0,1}n functions from the
asynchronous systems theory are called signals. The periodicity of a point of
the orbit of the signal x is defined and we give a note on the existence of
the prime period.

Keywords and phrases: binary signal, period, prime period.

MSC (2008): 94A12

The asynchronous systems are the models of the digital electrical circuits and
the ’nice’ functions representing their inputs and states are called signals.
Such systems are generated by Boolean functions that iterate like the
dynamical systems, but the iterations happen on some coordinates only, not on
all the coordinates (unlike the dynamical systems). In order to study their
periodicity, we need to study the periodicity of the (values of the) signals
first. Our present aim is to define and to characterize the periodicity and
the prime period of a point of the orbit of a signal.

Definition 1

The set B={0,1} is a field relative to ′⊕′,′⋅′, the modulo 2 sum and the
product. A linear space structure is induced on Bn,n≥1.

Notation 2

χA:R→B is the notation of the
characteristic function of the set A⊂R:∀t∈R,

χA(t)={1,ift∈A,0,otherwise.

Definition 3

The continuous time signals are the functions
x:R→Bn of the form ∀t∈R,

x(t)=μ⋅χ(−∞,t0)(t)⊕x(t0)⋅χ[t0,t1)(t)⊕...⊕x(tk)⋅χ[tk,tk+1)(t)⊕...

(1)

where μ∈Bn and tk∈R,k∈N is
strictly increasing and unbounded from above. Their set is denoted by
S(n).μ is usually denoted by x(−∞+0) and is called the
initial value of x.

Definition 4

The left limitx(t−0) of x(t) from (1) is by
definition the function ∀t∈R,

x(t−0)=μ⋅χ(−∞,t0](t)⊕x(t0)⋅χ(t0,t1](t)⊕...⊕x(tk)⋅χ(tk,tk+1](t)⊕...

(2)

Remark 5

The definition of x(t−0) does not depend on the choice of (tk) that is
not unique in (1); for any t′∈R, the existence
of x(t′−0) is used in applications under the form ∃ε>0,∀ξ∈(t′−ε,t′),x(ξ)=x(t′−0).

Definition 6

The set Or(x)={x(t)|t∈R} is called the orbit of x.

Notation 7

For x∈S(n) and μ∈Or(x), we denote

Txμ={t|t∈R,x(t)=μ}.

(3)

Definition 8

The point μ∈Or(x) is called a periodic point of x∈S(n)or of Or(x) if T>0,t′∈R exist such that

(−∞,t′]⊂Txx(−∞+0),

(4)

∀t∈Txμ∩[t′,∞),{t+zT|z∈Z}∩[t′,∞)⊂Tμx.

(5)

In this case T is called the period of μ and the least T like
above is called the prime period of μ.

Theorem 9

Let x∈S(n),μ=x(−∞+0),T>0 and the points
t0,t1∈R having the property that

t0<t1<t0+T,

(6)

(−∞,t0)∪[t1,t0+T)∪[t1+T,t0+2T)∪[t1+2T,t0+3T)∪...=Txμ

(7)

hold.

a) For any t′∈[t1−T,t0), the properties
(4), (5) are fulfilled and for any t′∉[t1−T,t0), at least one of the properties (4),
(5) is false.

We suppose now that t′∉[t1−T,t0). If t′<t1−T, we notice that max{t′,t0−T}<t1−T and that for
any t∈[max{t′,t0−T},t1−T), we have t∈Txμ∩[t′,∞) but

t+T∈{t+zT|z∈Z}∩[t′,∞)∩[t0,t1),

thus t+T∉Txμ and (5) is false. On the other
hand if t′≥t0, then x(t0)≠μ implies t0∉Txμ and consequently (4) is false.

b) The fact that t′′∈[t1−T′,t0) is
proved similarly with the statement t′∈[t1−T,t0) from
a): t′′≥t0 is in contradiction with (8) and
t′′<t1−T′ is in contradiction with (9).

We suppose now against all reason that (8), (9) are true
and T′<T. Let us note in the beginning that

max{t1,t0+T−T′}<min{t0+T,t1+T−T′}

is true, since all of t1<t0+T,t1<t1+T−T′,t0+T−T′<t0+T,t0+T−T′<t1+T−T′ are true. We
infer that any t∈[max{t1,t0+T−T′},min{t0+T,t1+T−T′}) fulfills t∈[t1,t0+T)⊂Txμ∩[t′′,∞) and

t0+T≤max{t1+T′,t0+T}≤t+T′<min{t0+T+T′,t1+T}≤t1+T

in other words t+T′∈{t+zT′|z∈Z}∩[t′′,∞), but t+T′∈[t0+T,t1+T), thus
t+T′∉Txμ, contradiction with (9).
We conclude that T′≥T.

Lemma 10

We suppose that the point μ∈Or(x) is periodic:
T>0,t′∈R exist such that (4), (5)
hold. If for t1<t2 we have [t1,t2)⊂Txμ∩[t′,∞), then ∀k≥1,[t1+kT,t2+kT)⊂Txμ.

Proof. Let k≥1 and t∈[t1+kT,t2+kT) be arbitrary. As
t−kT∈[t1,t2) and from the hypothesis t−kT∈Txμ∩[t′,∞), we have from (5) that

t∈{t−kT+zT|z∈Z}∩[t′,∞)⊂Txμ.

Theorem 11

We ask that x is not constant and let the point μ=x(−∞+0) be given, as well as T>0,t′∈R such that
(4), (5) hold. We define t0,t1∈R by
the requests

∀t<t0,x(t)=μ,

(10)

x(t0)≠μ,

(11)

t1<t0+T,

(12)

∀t∈[t1,t0+T),x(t)=x(t0+T−0),

(13)

x(t1−0)≠x(t1).

(14)

Then the following statements are true:

t1−T≤t′<t0<t1,

(15)

(−∞,t0)∪[t1,t0+T)∪[t1+T,t0+2T)∪[t1+2T,t0+3T)∪...⊂Txμ.

(16)

Proof. The fact that x is not constant assures the existence of t0 as defined
by (10), (11). On the other hand t1 as defined by
(12), (13), (14) exists itself, since if,
against all reason, we would have

∀t<t0+T,x(t)=x(t0+T−0),

(17)

then (10), (11), (17) would be contradictory. By
the comparison between (10), (11), (13),
(14) we infer t0≤t1. From (4), (10), (11) we get t′<t0.